Metric Transformations: Explained with Diagrams

[Muthumanimaran]

When I study about the transformation of coordinates, especially while defining gradient, curl, divergence and other vector integral theorem in different co-ordinate system, a concept called metric is defined and it is said to used for transform these operators in different co-ordinates, it is given as from a rectangular co-ordinate to any system ,it is given as the differential element dxi=hi qi (dqi is the differential element in other co-ordinate system) and hi is scaling factor, but my question is what is this scaling factor, the book I referred just defined this transformation equation and did not derived it, please explain me how this equation came from and explain with diagrams if possible.

 

[Orodruin]

The scale factor is (equivalently) the length of the vector
$$
\frac{\partial \vec r}{\partial q_i}, \quad \mbox{i.e.}\quad h_i = \left|\frac{\partial \vec r}{\partial q_i}\right|.
$$
These vectors can be used to form a normalised basis $\vec e_i$ according to
$$
\vec e_i = \frac{1}{h_i} \frac{\partial \vec r}{\partial q_i}.
$$
This is normally done in orthogonal coordinates in such a way that your basis becomes orthonormal.
The metric is a more general concept and defines distances in a general manifold. It can be used to determine the length of vectors.